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Full text of "Treatise On Analysis Vol-Ii"

(1    Definition ofmF.

Let us denote by <D ^^£($3) the set of all continuous functions F which
tend to 0 at infinity and for which there exists a bounded measure mF on Sg
satisfying ( for all x, y in A. We shall prove that <3> contains tfc($g)y
by means of the following three lemmas.

(1 0)   <D is an ideal in the algebra ^(Sg),

It is clear that <D is a vector space. Also if F e $ and G e ^(S^), it follows
from ( that

(GF) • mXty = (G*f) • mF = (jtf) - (G • mF),

so that GF e <3>, and we may take mGF = G • mF (because G is wF-integ-
rable (13.20.5)).

( For each function FeO, there exists only one bounded measure
mF satisfying ( for all x, y in A. The mapping Fh->/% is linear, and
mF ^ 0 whenever F §: 0.

To prove the uniqueness of WF , it is enough to show that the functions
of the form xj> (where x, y e A) form a total set in the Banach space ^(S^), or
again that the functions of the form A + jep (A e C) form a total set in the
Banach space ^C(S^)- Now, from the fact that A is an algebra and x\-+x is an
algebra homomorphism, it follows that the set B of these functions is a sub-
algebra of #c(Sp» containing the constants. This subalgebra separates the
points of S^. For if ft, #2 are two distinct points of Sg, then by virtue of (ii)
there exists x e A such that xfa) ^ x(x2)- Hence either j£2(ft) ^ .*2fe) or
•#3fe) 7* ^3(X2). On the other hand, if S; = S^ u {0}, then for each x ¥* 0 there
exists x e A such that x(%) ^ 0, and therefore x2(%) ^ 0, which proves our
assertion. Finally, since 5 = (x*)~ (15.9.1), the conjugate of each function in B
belongs to B. Hence we may apply the Stone-Weierstrass theorem (7.3.2),
which establishes the uniqueness of mF . The linearity of the mapping FH-»/%
follows immediately from the uniqueness just established. It remains now to
show that the relation F J2; 0 implies that mF > 0. Let G JS> 0 be a function
belonging to ^fR(S^). By virtue of (ii), there exists a sequence (*„) of elements
of A such that the functions xn converge uniformly on S^ to the function
G1/2. The functions xnxn therefore converge uniformly on S^ to G, and hence,
by virtue of (

= lim t\

n-+oo Jrmation is an isometry of stg onto ^c(X(*aQ) (15.4.14); the